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Petar Veličković

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14 papers

1 author row

IJCAI Conference 2024 Conference Paper

Finding Increasingly Large Extremal Graphs with AlphaZero and Tabu Search

Abbas Mehrabian
Ankit Anand
Hyunjik Kim
Nicolas Sonnerat
Matej Balog
Gheorghe Comanici
Tudor Berariu
Andrew Lee

This work proposes a new learning-to-search benchmark and uses AI to discover new mathematical knowledge related to an open conjecture of Erdos (1975) in extremal graph theory. The problem is to find graphs with a given size (number of nodes) that maximize the number of edges without having 3- or 4-cycles. We formulate this as a sequential decision-making problem and compare AlphaZero, a neural network-guided tree search, with tabu search, a heuristic local search method. Using either method, by introducing a curriculum---jump-starting the search for larger graphs using good graphs found at smaller sizes---we improve the state-of-the-art lower bounds for several sizes. We also propose a flexible graph-generation environment and a permutation-invariant network architecture for learning to search in the space of graphs.

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TMLR Journal 2024 Journal Article

How does over-squashing affect the power of GNNs?

Francesco Di Giovanni
T. Konstantin Rusch
Michael Bronstein
Andreea Deac
Marc Lackenby
Siddhartha Mishra
Petar Veličković

Graph Neural Networks (GNNs) are the state-of-the-art model for machine learning on graph-structured data. The most popular class of GNNs operate by exchanging information between adjacent nodes, and are known as Message Passing Neural Networks (MPNNs). While understanding the expressive power of MPNNs is a key question, existing results typically consider settings with uninformative node features. In this paper, we provide a rigorous analysis to determine which function classes of node features can be learned by an MPNN of a given capacity. We do so by measuring the level of *pairwise interactions* between nodes that MPNNs allow for. This measure provides a novel quantitative characterization of the so-called over-squashing effect, which is observed to occur when a large volume of messages is aggregated into fixed-size vectors. Using our measure, we prove that, to guarantee sufficient communication between pairs of nodes, the capacity of the MPNN must be large enough, depending on properties of the input graph structure, such as commute times. For many relevant scenarios, our analysis results in impossibility statements in practice, showing that *over-squashing hinders the expressive power of MPNNs*. Our theory also holds for geometric graphs and hence extends to equivariant MPNNs on point clouds. We validate our analysis through extensive controlled experiments and ablation studies.

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NeurIPS Conference 2024 Conference Paper

Transformers need glasses! Information over-squashing in language tasks

Federico Barbero
Andrea Banino
Steven Kapturowski
Dharshan Kumaran
João G. Araújo
Alex Vitvitskyi
Razvan Pascanu
Petar Veličković

We study how information propagates in decoder-only Transformers, which are the architectural foundation of most existing frontier large language models (LLMs). We rely on a theoretical signal propagation analysis---specifically, we analyse the representations of the last token in the final layer of the Transformer, as this is the representation used for next-token prediction. Our analysis reveals a representational collapse phenomenon: we prove that certain distinct pairs of inputs to the Transformer can yield arbitrarily close representations in the final token. This effect is exacerbated by the low-precision floating-point formats frequently used in modern LLMs. As a result, the model is provably unable to respond to these sequences in different ways---leading to errors in, e. g. , tasks involving counting or copying. Further, we show that decoder-only Transformer language models can lose sensitivity to specific tokens in the input, which relates to the well-known phenomenon of over-squashing in graph neural networks. We provide empirical evidence supporting our claims on contemporary LLMs. Our theory points to simple solutions towards ameliorating these issues.

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NeurIPS Conference 2023 Conference Paper

Affinity-Aware Graph Networks

Ameya Velingker
Ali Sinop
Ira Ktena
Petar Veličković
Sreenivas Gollapudi

Graph Neural Networks (GNNs) have emerged as a powerful technique for learning on relational data. Owing to the relatively limited number of message passing steps they perform—and hence a smaller receptive field—there has been significant interest in improving their expressivity by incorporating structural aspects of the underlying graph. In this paper, we explore the use of affinity measures as features in graph neural networks, in particular measures arising from random walks, including effective resistance, hitting and commute times. We propose message passing networks based on these features and evaluate their performance on a variety of node and graph property prediction tasks. Our architecture has low computational complexity, while our features are invariant to the permutations of the underlying graph. The measures we compute allow the network to exploit the connectivity properties of the graph, thereby allowing us to outperform relevant benchmarks for a wide variety of tasks, often with significantly fewer message passing steps. On one of the largest publicly available graph regression datasets, OGB-LSC-PCQM4Mv1, we obtain the best known single-model validation MAE at the time of writing.

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JMLR Journal 2023 Journal Article

Combinatorial Optimization and Reasoning with Graph Neural Networks

Quentin Cappart
Didier Chételat
Elias B. Khalil
Andrea Lodi
Christopher Morris
Petar Veličković

Combinatorial optimization is a well-established area in operations research and computer science. Until recently, its methods have focused on solving problem instances in isolation, ignoring that they often stem from related data distributions in practice. However, recent years have seen a surge of interest in using machine learning, especially graph neural networks, as a key building block for combinatorial tasks, either directly as solvers or by enhancing exact solvers. The inductive bias of GNNs effectively encodes combinatorial and relational input due to their invariance to permutations and awareness of input sparsity. This paper presents a conceptual review of recent key advancements in this emerging field, aiming at optimization and machine learning researchers. [abs] [ pdf ][ bib ] &copy JMLR 2023. ( edit, beta )

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AAAI Conference 2022 Conference Paper

Algorithmic Concept-Based Explainable Reasoning

Dobrik Georgiev
Pietro Barbiero
Dmitry Kazhdan
Petar Veličković
Pietro Lió

Recent research on graph neural network (GNN) models successfully applied GNNs to classical graph algorithms and combinatorial optimisation problems. This has numerous benefits, such as allowing applications of algorithms when preconditions are not satisfied, or reusing learned models when sufficient training data is not available or can’t be generated. Unfortunately, a key hindrance of these approaches is their lack of explainability, since GNNs are black-box models that cannot be interpreted directly. In this work, we address this limitation by applying existing work on concept-based explanations to GNN models. We introduce concept-bottleneck GNNs, which rely on a modification to the GNN readout mechanism. Using three case studies we demonstrate that: (i) our proposed model is capable of accurately learning concepts and extracting propositional formulas based on the learned concepts for each target class; (ii) our concept-based GNN models achieve comparative performance with state-of-the-art models; (iii) we can derive global graph concepts, without explicitly providing any supervision on graph-level concepts.

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NeurIPS Conference 2022 Conference Paper

Graph Neural Networks are Dynamic Programmers

Andrew J Dudzik
Petar Veličković

Recent advances in neural algorithmic reasoning with graph neural networks (GNNs) are propped up by the notion of algorithmic alignment. Broadly, a neural network will be better at learning to execute a reasoning task (in terms of sample complexity) if its individual components align well with the target algorithm. Specifically, GNNs are claimed to align with dynamic programming (DP), a general problem-solving strategy which expresses many polynomial-time algorithms. However, has this alignment truly been demonstrated and theoretically quantified? Here we show, using methods from category theory and abstract algebra, that there exists an intricate connection between GNNs and DP, going well beyond the initial observations over individual algorithms such as Bellman-Ford. Exposing this connection, we easily verify several prior findings in the literature, produce better-grounded GNN architectures for edge-centric tasks, and demonstrate empirical results on the CLRS algorithmic reasoning benchmark. We hope our exposition will serve as a foundation for building stronger algorithmically aligned GNNs.

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NeurIPS Conference 2022 Conference Paper

Learning to Configure Computer Networks with Neural Algorithmic Reasoning

Luca Beurer-Kellner
Martin Vechev
Laurent Vanbever
Petar Veličković

We present a new method for scaling automatic configuration of computer networks. The key idea is to relax the computationally hard search problem of finding a configuration that satisfies a given specification into an approximate objective amenable to learning-based techniques. Based on this idea, we train a neural algorithmic model which learns to generate configurations likely to (fully or partially) satisfy a given specification under existing routing protocols. By relaxing the rigid satisfaction guarantees, our approach (i) enables greater flexibility: it is protocol-agnostic, enables cross-protocol reasoning, and does not depend on hardcoded rules; and (ii) finds configurations for much larger computer networks than previously possible. Our learned synthesizer is up to 490x faster than state-of-the-art SMT-based methods, while producing configurations which on average satisfy more than 93% of the provided requirements.

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IJCAI Conference 2021 Conference Paper

Combinatorial Optimization and Reasoning with Graph Neural Networks

Quentin Cappart
Didier Chételat
Elias B. Khalil
Andrea Lodi
Christopher Morris
Petar Veličković

Combinatorial optimization is a well-established area in operations research and computer science. Until recently, its methods have mostly focused on solving problem instances in isolation, ignoring the fact that they often stem from related data distributions in practice. However, recent years have seen a surge of interest in using machine learning, especially graph neural networks, as a key building block for combinatorial tasks, either directly as solvers or by enhancing the former. This paper presents a conceptual review of recent key advancements in this emerging field, aiming at researchers in both optimization and machine learning.

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NeurIPS Conference 2021 Conference Paper

How to transfer algorithmic reasoning knowledge to learn new algorithms?

Louis-Pascal Xhonneux
Andreea-Ioana Deac
Petar Veličković
Jian Tang

Learning to execute algorithms is a fundamental problem that has been widely studied. Prior work (Veličković et al. , 2019) has shown that to enable systematic generalisation on graph algorithms it is critical to have access to the intermediate steps of the program/algorithm. In many reasoning tasks, where algorithmic-style reasoning is important, we only have access to the input and output examples. Thus, inspired by the success of pre-training on similar tasks or data in Natural Language Processing (NLP) and Computer vision, we set out to study how we can transfer algorithmic reasoning knowledge. Specifically, we investigate how we can use algorithms for which we have access to the execution trace to learn to solve similar tasks for which we do not. We investigate two major classes of graph algorithms, parallel algorithms such as breadth-first search and Bellman-Ford and sequential greedy algorithms such as Prims and Dijkstra. Due to the fundamental differences between algorithmic reasoning knowledge and feature extractors such as used in Computer vision or NLP, we hypothesis that standard transfer techniques will not be sufficient to achieve systematic generalisation. To investigate this empirically we create a dataset including 9 algorithms and 3 different graph types. We validate this empirically and show how instead multi-task learning can be used to achieve the transfer of algorithmic reasoning knowledge.

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NeurIPS Conference 2021 Conference Paper

Neural Algorithmic Reasoners are Implicit Planners

Andreea-Ioana Deac
Petar Veličković
Ognjen Milinkovic
Pierre-Luc Bacon
Jian Tang
Mladen Nikolic

Implicit planning has emerged as an elegant technique for combining learned models of the world with end-to-end model-free reinforcement learning. We study the class of implicit planners inspired by value iteration, an algorithm that is guaranteed to yield perfect policies in fully-specified tabular environments. We find that prior approaches either assume that the environment is provided in such a tabular form---which is highly restrictive---or infer "local neighbourhoods" of states to run value iteration over---for which we discover an algorithmic bottleneck effect. This effect is caused by explicitly running the planning algorithm based on scalar predictions in every state, which can be harmful to data efficiency if such scalars are improperly predicted. We propose eXecuted Latent Value Iteration Networks (XLVINs), which alleviate the above limitations. Our method performs all planning computations in a high-dimensional latent space, breaking the algorithmic bottleneck. It maintains alignment with value iteration by carefully leveraging neural graph-algorithmic reasoning and contrastive self-supervised learning. Across seven low-data settings---including classical control, navigation and Atari---XLVINs provide significant improvements to data efficiency against value iteration-based implicit planners, as well as relevant model-free baselines. Lastly, we empirically verify that XLVINs can closely align with value iteration.

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NeurIPS Conference 2021 Conference Paper

Neural Distance Embeddings for Biological Sequences

Gabriele Corso
Zhitao Ying
Michal Pándy
Petar Veličković
Jure Leskovec
Pietro Liò

The development of data-dependent heuristics and representations for biological sequences that reflect their evolutionary distance is critical for large-scale biological research. However, popular machine learning approaches, based on continuous Euclidean spaces, have struggled with the discrete combinatorial formulation of the edit distance that models evolution and the hierarchical relationship that characterises real-world datasets. We present Neural Distance Embeddings (NeuroSEED), a general framework to embed sequences in geometric vector spaces, and illustrate the effectiveness of the hyperbolic space that captures the hierarchical structure and provides an average 38% reduction in embedding RMSE against the best competing geometry. The capacity of the framework and the significance of these improvements are then demonstrated devising supervised and unsupervised NeuroSEED approaches to multiple core tasks in bioinformatics. Benchmarked with common baselines, the proposed approaches display significant accuracy and/or runtime improvements on real-world datasets. As an example for hierarchical clustering, the proposed pretrained and from-scratch methods match the quality of competing baselines with 30x and 15x runtime reduction, respectively.

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NeurIPS Conference 2020 Conference Paper

Pointer Graph Networks

Petar Veličković
Lars Buesing
Matthew Overlan
Razvan Pascanu
Oriol Vinyals
Charles Blundell

Graph neural networks (GNNs) are typically applied to static graphs that are assumed to be known upfront. This static input structure is often informed purely by insight of the machine learning practitioner, and might not be optimal for the actual task the GNN is solving. In absence of reliable domain expertise, one might resort to inferring the latent graph structure, which is often difficult due to the vast search space of possible graphs. Here we introduce Pointer Graph Networks (PGNs) which augment sets or graphs with additional inferred edges for improved model generalisation ability. PGNs allow each node to dynamically point to another node, followed by message passing over these pointers. The sparsity of this adaptable graph structure makes learning tractable while still being sufficiently expressive to simulate complex algorithms. Critically, the pointing mechanism is directly supervised to model long-term sequences of operations on classical data structures, incorporating useful structural inductive biases from theoretical computer science. Qualitatively, we demonstrate that PGNs can learn parallelisable variants of pointer-based data structures, namely disjoint set unions and link/cut trees. PGNs generalise out-of-distribution to 5x larger test inputs on dynamic graph connectivity tasks, outperforming unrestricted GNNs and Deep Sets.

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NeurIPS Conference 2020 Conference Paper

Principal Neighbourhood Aggregation for Graph Nets

Gabriele Corso
Luca Cavalleri
Dominique Beaini
Pietro Liò
Petar Veličković

Graph Neural Networks (GNNs) have been shown to be effective models for different predictive tasks on graph-structured data. Recent work on their expressive power has focused on isomorphism tasks and countable feature spaces. We extend this theoretical framework to include continuous features---which occur regularly in real-world input domains and within the hidden layers of GNNs---and we demonstrate the requirement for multiple aggregation functions in this context. Accordingly, we propose Principal Neighbourhood Aggregation (PNA), a novel architecture combining multiple aggregators with degree-scalers (which generalize the sum aggregator). Finally, we compare the capacity of different models to capture and exploit the graph structure via a novel benchmark containing multiple tasks taken from classical graph theory, alongside existing benchmarks from real-world domains, all of which demonstrate the strength of our model. With this work we hope to steer some of the GNN research towards new aggregation methods which we believe are essential in the search for powerful and robust models.

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